Horst Faas did some truly remarkable work, but for the sake of clarity, neither of the photos was actually taken by him. The top photo of the prisoner execution was done by Eddie Adams; the bottom one of the running girl was by Nick Ut, a Vietnamese. Both Adams, an American, and Ut worked for Faas out of the AP's Saigon bureau.

There's also an interesting backstory, for those not aware of it, to the Adams photo that provides a bit more context to what was going on at the time it was taken: